Problem 72 - Intermediate Algebra

Question

Perform each division. \(\frac{2 x^{4}+3 x^{3}+3 x^{2}-5 x-3}{2 x^{2}-x-1}\)

Step-by-Step Solution

Answer

The result is \(x^2 + 2x + 3 - \frac{x}{2x^2 - x - 1}\).

Step 1: Set Up the Long Division

Write the dividend \(2x^4 + 3x^3 + 3x^2 - 5x - 3\) inside the division symbol and the divisor \(2x^2 - x - 1\) outside. This sets up the polynomial long division.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(2x^4\) by the leading term of the divisor \(2x^2\). The result is \(x^2\). Write \(x^2\) above the division symbol.

Step 3: Multiply and Subtract

Multiply \(x^2\) by the entire divisor \(2x^2 - x - 1\) to get \(2x^4 - x^3 - x^2\). Subtract this from the original dividend to get a new polynomial \(4x^3 + 4x^2 - 5x - 3\).

Step 4: Repeat the Process

Divide the new leading term \(4x^3\) by \(2x^2\) to get \(2x\). Multiply and subtract \((2x)(2x^2 - x - 1)\) from \(4x^3 + 4x^2 - 5x - 3\) to get \(6x^2 - 3x - 3\).

Step 5: Divide Again

Divide the new leading term \(6x^2\) by \(2x^2\) to get \(3\). Write \(3\) above the division symbol. Multiply and subtract \((3)(2x^2 - x - 1)\) from \(6x^2 - 3x - 3\) to end with \(-x\).

Step 6: Write Down the Remainder

Since you can't divide further, write \(-x\) as the remainder. The result of the division is \(x^2 + 2x + 3\) with remainder \(-x\): \(x^2 + 2x + 3 + \frac{-x}{2x^2 - x - 1}\).

Key Concepts

Long Division

Long division is a method used to divide polynomials, similar to the long division you might remember from dividing numbers. The key idea is to work through the dividend (the polynomial you're dividing) one term at a time using the divisor (the polynomial you're dividing by). This approach helps organize the process and maintain clarity throughout the division.

Start by setting up the problem: place the dividend under the long division symbol and the divisor outside. This setup is crucial as it defines the framework within which you will perform the division. For example, in the exercise with the dividend \(2x^4 + 3x^3 + 3x^2 - 5x - 3\) and divisor \(2x^2 - x - 1\), setting up clearly will guide you through the subsequent steps efficiently.

• Identify the leading terms first to begin each step.
• Label each part of the division process above the division symbol.
• Carefully subtract to find the next new dividend term.

Using long division requires patience and careful computation but ensures accuracy in polynomial division.

Remainder

In polynomial division, the remainder is what you have left when you have divided the polynomial as much as possible but cannot continue further. It is a smaller polynomial with a degree less than the divisor. The presence of a remainder indicates that the division does not result in a whole number of the divisor.

When performing the division, you subtract the result of multiplying the partial quotient by the divisor from the current dividend term by term. If, at the end of the process, there is any remaining polynomial that can't be divided by the divisor, this becomes your remainder. In our example, after completing the division steps, the remainder is \(-x\).

• The remainder is often expressed as a fraction where it is divided by the original divisor.
• In this example, the final result reads as \(x^2 + 2x + 3 + \frac{-x}{2x^2 - x - 1}\).
• A remainder of zero indicates complete divisibility.

Understanding the remainder helps verify the completeness and accuracy of the polynomial division.

Leading Terms

The leading term of a polynomial is the term with the highest degree, and it plays a critical role in polynomial division. Focusing on the leading terms first helps simplify and guide the division process, making it more manageable.

You start the division by dividing the leading term of the dividend by the leading term of the divisor. In the exercise given, this first involved dividing \(2x^4\) by \(2x^2\), resulting in \(x^2\). This value, \(x^2\), is placed at the top of the division structure.

• Repeat this step with each new divisor formed after subtraction.
• The leading term is crucial for determining the correct multiplier for each division step.
• The process continues until the remaining terms have a degree less than the divisor.

Focusing on leading terms avoids distraction by lesser degree terms and ensures systematic division.

Polynomial Operations

Polynomial operations like addition, subtraction, multiplication, and division are fundamental in algebra. They allow us to manipulate polynomial expressions, making this suite of skills essential for problem-solving in mathematics.

When dividing polynomials, especially through long division, these operations are frequently used:

• Subtraction: After multiplying the divisor by the term obtained by division, subtract this from the current polynomial.
• Multiplication: As seen in the exercise, after obtaining a new term from dividing the leading terms, multiply it by the entire divisor.
• Addition: While not directly used in the steps of this example, adding polynomials often helps in simplifying and combining terms post-division.

Mastering these operations provides a toolkit for handling complex polynomial expressions and prepares for more advanced topics like factoring and solving polynomial equations.